Pagina:Gauss, Carl Friedrich - Werke (1870).djvu/77

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moduli qui sunt numerorum primum potestates.

quare quisque terminus $(\alpha +hp^{\mu })^{t-1}\!$, $(\alpha +hp^{\mu })^{t-2}\alpha$ etc. erit $\textstyle \equiv \alpha ^{t-1}{\pmod {p^{2}}}$, adeoque omnium summa $\textstyle \equiv t\alpha ^{t-1}{\pmod {p^{2}}}$ sive formae $t\alpha ^{t-1}+Vp^{2}$ denotante $V$ numerum quemcunque. Hinc $(\alpha +hp^{\mu })^{t}-\alpha ^{t}$ erit formae $\alpha ^{t-1}hp^{\mu }t+Vhp^{\mu +2}$, i. e. $\textstyle \equiv \alpha ^{t-1}hp^{\mu }t{\pmod {p^{\mu +2}}}$ et $\textstyle \equiv 0{\pmod {p^{\mu +1}}}$ Pro hoc itaque casu theorema est demonstratum.

Iam si theorema pro aliis ipsius $\nu$ valoribus verum non esset, manente etiamnum $\mu >1$, limes aliquis necessario daretur, usque ad quem theorema semper verum foret, ultra vero falsum. Sit minimus valor ipsius $\nu$, pro quo falsum est $=\varphi$, unde facile perspicitur, si $t$ per $p^{\varphi -1}\!$ non autem per $p^{\varphi }\!$ fuerit divisibilis, theorema adhuc verum esse, at si loco ipsius $t$ substituatur $tp$, falsum. Habemus itaque $\textstyle (\alpha +hp^{\mu })^{t}\equiv \alpha ^{t}+\alpha ^{t-1}hp^{\mu }t{\pmod {p^{\mu +\varphi }}}$ sive $=\alpha ^{t}+\alpha ^{t-1}hp^{\mu }t+up^{\mu +\varphi }$ denotante $u$ numerum integrum. At quia pro $\nu =1$ theorema iam est demonstratum, erit $\textstyle (\alpha ^{t}+\alpha ^{t-1}hp^{\mu }t+up^{\mu +\varphi })^{p}\equiv$ $\textstyle \alpha ^{tp}+\alpha ^{tp-1}hp^{\mu +1}t+\alpha ^{tp-t}up^{\mu +\varphi +1}{\pmod {p^{\mu +\varphi +1}}}$ adeoque etiam $\textstyle (\alpha +hp^{\mu })^{tp}\equiv \alpha ^{tp}+\alpha ^{tp-1}hp^{\mu }tp{\pmod {p^{\mu +\varphi +1}}}$ i. e. theorema etiam verum, si loco ipsius $t$ substituitur $tp$, i. e. etiam pro $\nu =\varphi$, contra hypothesin. Unde manifestum pro omnibus ipsius $\nu$ valoribus theorema verum esse.

87.

Superest casus ubi $\mu =1$. Per methodum prorsus similem ei qua in art. praec. usi sumus, sine adiumento theorematis binomialis demonstrari potest, esse ${\begin{alignedat}{1}\textstyle (\alpha +&hp)^{t-1}\equiv &\ \alpha ^{t-1}+\alpha ^{t-2}(t-1)hp{\pmod {p^{2}}}\\\textstyle \alpha (\alpha +&hp)^{t-2}\equiv &\ \alpha ^{t-1}+\alpha ^{t-2}(t-2)hp\\\textstyle \alpha \alpha (\alpha +&hp)^{t-3}\equiv &\ \alpha ^{t-1}+\alpha ^{t-2}(t-3)hp\\\textstyle &\mathrm {etc.} \end{alignedat}}$ unde aggregatum erit (quia partium raultitudo $=t$)